Post on 13-Sep-2020
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Lecture 12Nash Equilibrium
Jitesh H. Panchal
ME 597: Decision Making for Engineering Systems Design
Design Engineering Lab @ Purdue (DELP)School of Mechanical Engineering
Purdue University, West Lafayette, INhttp://engineering.purdue.edu/delp
October 30, 2014
c©Jitesh H. Panchal Lecture 12 1 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Lecture Outline
1 Nash Equilibrium
2 Relation between IEDS and Nash equilibrium
3 Application: Cournot Duopoly
4 Application: The Commons Problem
Dutta, P.K. (1999). Strategies and Games: Theory and Practice. Cambridge, MA, The MIT Press. Chapters 5 and 6.
c©Jitesh H. Panchal Lecture 12 2 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Best Response
Best Response
A strategy s∗i is a best response to a strategy vector s∗
−i of the other players if
πi (s∗i , s
∗−i ) ≥ πi (si , s∗
−i ), for all si
s∗i is a “dominant strategy” in a very weak sense.
c©Jitesh H. Panchal Lecture 12 3 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium
Nash Equilibrium
The strategy vector s∗ = s∗1 , s
∗2 , . . . , s
∗N is a Nash equilibrium if
πi (s∗i , s
∗−i ) ≥ πi (si , s∗
−i ), for all si and all i
Requirements of Nash equilibrium:
Each player must be playing a best response against a conjecture.
The conjectures must be correct.
c©Jitesh H. Panchal Lecture 12 4 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example of a two-player scenario
Two players: 1 and 2
Strategies:1 {a1 and a2} for player 12 {b1 and b2} for player 2
(a2, b1) is a Nash equilibrium if and only if
π1(a2, b1) ≥ π1(a1, b1)
π2(a2, b1) ≥ π2(a2, b2)
c©Jitesh H. Panchal Lecture 12 5 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example 1
Battle of Sexes
Husband / Wife Football (F) OperaFootball (F) 3, 1 0, 0
Opera (O) 0, 0 1, 3
c©Jitesh H. Panchal Lecture 12 6 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example 2
Prisoner’s Dilemma
1 / 2 Confess Not ConfessConfess 0, 0 7,−2
Not Confess −2, 7 5, 5
c©Jitesh H. Panchal Lecture 12 7 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example 3
Bertrand Pricing
Firm 1 / Firm 2 High (H) Medium (M) Low (L)High (H) 6, 6 0, 10 0, 8
Medium (M) 10, 0 5, 5 0, 8Low (L) 8, 0 8, 0 4, 4
c©Jitesh H. Panchal Lecture 12 8 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example 4
The Odd Couple
Felix / Oscar 3 hours 6 hours 9 hours3 hours −13,−8 −1,−4 7,−46 hours −4,−1 4,−1 4,−49 hours 1, 2 1,−1 1,−4
c©Jitesh H. Panchal Lecture 12 9 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Nash Equilibrium - Example 5
Coordination game
1 / 2 7:30 10:307:30 1, 1 0, 0
10:30 0, 0 0, 0
c©Jitesh H. Panchal Lecture 12 10 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Motivation for Nash Equilibrium
Scenarios under which the two requirements (players playing best responseagainst conjectures, and conjectures being correct) may be appropriate:
1 Play prescription2 Preplay communication3 Rational introspection4 Focal point5 Trial and error
c©Jitesh H. Panchal Lecture 12 11 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Relationship between IEDS and Nash equilibrium
Proposition
Consider any game in which there is an outcome to IEDS. It must be the casethat this outcome is a Nash equilibrium.
However, not every Nash equilibrium can be obtained as the outcome toIEDS.
c©Jitesh H. Panchal Lecture 12 12 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Relationship between IEDS and Nash equilibrium - Illustration
Consider a 2-player 3-strategy game.
A / B b1 b2 b3
a1
a2
a3
Suppose that {a2, b3} is the IEDS solution.
For Nash equilibrium, need to show that:
a2 � a1, a2 � a3 when played against b3
b3 � b1, b3 � b2 when played against a2
c©Jitesh H. Panchal Lecture 12 13 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Application: Cournot Duopoly
Extreme types of markets:
Monopoly (single firm)
Perfect competition (infinitely many firms)
Few firms in a given market:
Automobile market : 3 domestic and 10 foreign manufacturers
Aircraft manufacturers: 1 domestic manufacturer and 1 foreignmanufacturer
World oil market : 10 manufacturing nations account for 80% of oilproduction
c©Jitesh H. Panchal Lecture 12 14 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
The Basic Cournot Duopoly Model
Firms compete in a market for ahomogenous product (single demandcurve).
Q = α− βP
where α > 0, β > 0, and Q = Q1 + Q2
Cost: C1 = cQ1 and C2 = cQ2
Question
How much should each firm produce?
Quantity (Q)
Price (P)
0 10
10
Figure: 6.1 on Page 77
c©Jitesh H. Panchal Lecture 12 15 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
How much should each firm produce?
1 Make a conjecture about other firm’s production.2 Determine the quantity to produce.
c©Jitesh H. Panchal Lecture 12 16 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Cournot Nash Equilibrium
Step 2
Make a conjecture about other firm’s production (say Q2).
The price is:
P =α
β− Q1 + Q2
β
P = a− b(Q1 + Q2)
where a =α
β, b =
1β
Step 2
Determine the quantity to produce.
MaxQ1 [a− b(Q1 + Q2)− c]Q1
c©Jitesh H. Panchal Lecture 12 17 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Cournot Nash Equilibrium: Best Response Functions
R1(Q2) =
a− c − bQ2
2bif Q2 ≤
a− cb
;
0 if Q2 >a− c
b.
R2(Q1) =
a− c − bQ1
2bif Q1 ≤
a− cb
;
0 if Q1 >a− c
b.
Solving these equations for Q1 and Q2, we get:
Q1 = Q2 = Q∗ =a− c
3b
P∗ =a + 2c
3
c©Jitesh H. Panchal Lecture 12 18 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Cournot Duopoly: Cartel Solution
If the two firms operate as a cartel, they coordinate their production decisionsto maximize their joint profits.
MaxQ1,Q2 [a− b(Q1 + Q2)− c][Q1 + Q2]
Solution:Q1 = Q2 = Q =
a− c4b
P =a + c
2
Per-Firm Profit =(a− c)2
8b
c©Jitesh H. Panchal Lecture 12 19 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Generalization: Cournot Oligopoly
For N firms,Q = Q1 + Q2 + · · ·+ QN
Best reply correspondence:
R1¯(Qi ) =
a− c − (N − 1)bQi
2b
Nash equilibrium quantities:
Q∗i =
a− c(N + 1)b
Price:P∗ =
aN + 1
+Nc
N + 1
c©Jitesh H. Panchal Lecture 12 20 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Generalization: Stackelberg Model
Suppose Firm 1 decides on its quantity before Firm 2 (i.e., Firm 2 knows Firm1’s quantity).
1 Firm 1 knows that Firm 2 will play a best response2 So, Firm 1 should choose Q1 knowing that Q2 = R2(Q1)
MaxQ1{a− b[Q1 + R2(Q1)]− c}Q1
Solving this we, get:
Q1 =a− c
2b
Q2 =a− c
4bNote: Firm 1’s profits are higher in the Stackelberg solution than in the Nashequilibrium.
c©Jitesh H. Panchal Lecture 12 21 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
The Commons Problem
Suppose that there is a common property resource of size y > 0
Each player can consume a non-negative amount, c1 or c2
Consider two time periods:1 Decide how much to consume in the first period2 Decide how much to consume from the available quantity: y − (c1 + c2)
c©Jitesh H. Panchal Lecture 12 22 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
The Commons Problem: Nash equilibrium
Assume: Utility from consumption: log(c1) and log(c2)
Player 1’s best response problem:
Maxc1 log(c1) + logy − (c1 + c2)
2
Solving this,
R1(c2) =y − c2
2Similarly,
R2(c1) =y − c1
2
The Nash equilibrium is: c∗1 = c∗
2 =y3
c©Jitesh H. Panchal Lecture 12 23 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
The Commons Problem: Social Optimality
Definition (Social optimality)
A pattern of comsumption, c1, c2 is socially optimal if it maximizes the sum ofthe two players’ utility, that is, if it solves the following problem:
Maxc1,c2 = log(c1) + log(c2) + 2 logy − (c1 + c2)
2
Socially optimal solution:c1 = c2 =
y4
c©Jitesh H. Panchal Lecture 12 24 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
The Commons Problem: For a Large Population
Nash equilibrium:c1 = c2 = · · · =
yN + 1
Socially optimal solution:
c1 = c2 = · · · =y
2N
Key Question
How can we balance the private desire for utility or profits against the socialimperative of sustainable resource use?
c©Jitesh H. Panchal Lecture 12 25 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
Summary
1 Nash Equilibrium
2 Relation between IEDS and Nash equilibrium
3 Application: Cournot Duopoly
4 Application: The Commons Problem
c©Jitesh H. Panchal Lecture 12 26 / 27
Nash EquilibriumRelation between IEDS and Nash equilibrium
Application: Cournot DuopolyApplication: The Commons Problem
References
1 Dutta, P.K. (1999). Strategies and Games: Theory and Practice.Cambridge, MA, The MIT Press. Chapters 5 and 6.
2 Keeney, R. L. and H. Raiffa (1993). Decisions with Multiple Objectives:Preferences and Value Tradeoffs. Cambridge, UK, Cambridge UniversityPress.
3 Clemen, R. T. (1996). Making Hard Decisions: An Introduction toDecision Analysis. Belmont, CA, Wadsworth Publishing Company.
4 Simon, H.A., A Behavioral Model of Rational Choice, Vol. 69, No. 1, pp.99-118.
5 Stanovich, K.E. (1999) Who is Rational? Studies of IndividualDifferences in Reasoning, Psychology Press, Mahwah, NJ.
c©Jitesh H. Panchal Lecture 12 27 / 27
THANK YOU!
c©Jitesh H. Panchal Lecture 12 1 / 1